PHY 2048CGeneral Physics I with lab

Spring 2011CRNs 11154, 11161 & 11165

Dr. Derrick BoucherAssoc. Prof. of Physics

Session 2, Chapter 3

Chapter 3

•READ IT

•Work out example problems

•Only a couple LON-CAPA questions come

directly from this

•But, concepts are essential throughout

this course and the next (PHY 2049)

Chapter 3 Practice Problems

Chap 3: 3, 7, 9, 11, 13, 15, 19, 25, 27

Unless otherwise indicated, all practice material is from the “Exercises and Problems” section at the end of the chapter. (Not “Questions.”)

• The need for vectors• Graphical representation• Mathematical formulation: components and unit vectors• Vector algebra with components

Outline

Chapter 3. Reading Review Chapter 3. Reading Review QuestionsQuestions

Starting next week, questions like the following

could be quiz questions.

PRSClicker

Questions

What is a vector?

A. A quantity having both size and directionB. The rate of change of velocityC. A number defined by an angle and a

magnitudeD. The difference between initial and final

displacementE. None of the above

What is the name of the quantity represented as ?

A. Eye-hatB. Invariant magnitudeC. Integral of motionD. Unit vector in x-directionE. Length of the horizontal axis

i

To decompose a vector means

A. to break it into several smaller vectors.B. to break it apart into scalars.C. to break it into pieces parallel to the axes.D. to place it at the origin.E. This topic was not discussed in Chapter 3.

Chapter 3. Basic Content and Chapter 3. Basic Content and ExamplesExamples

EXAMPLE 3.2 Velocity and displacement

QUESTION:

EXAMPLE 3.2 Velocity and displacement

EXAMPLE 3.2 Velocity and displacement

EXAMPLE 3.2 Velocity and displacement

EXAMPLE 3.2 Velocity and displacement

Components

•The x and y components of a vector tell us how much

the vector lies along the x and y axes, respectively.

•To calculate components of vectors you need a good

diagram and some simple trigonometry.

•There are no shortcuts here. The ONLY way to work

with vectors are via components.

Example problem Chapter 3 #6 (p. 87)

Unit vectors

i j k

•Unit vectors are a fancy way to say “thattaway” in mathematical notation.•Whichaway? Well, “north”, “west”, “up”, “right” are all examples of specific directions. These are, in a sense, unit vectors.•“West” doesn’t say how far, just what direction.•“Positive x direction”, etc. are also unit vector concepts.

For the positive x, y and z directions, our text uses “eye hat”, “jay hat” and “kay hat.”

Unit vectorsX Y Z

i j k

x y z

Some math and science texts use these symbols

Unit vectors

mis ˆ14

TjB ˆ073.0

mkh ˆ10287.1 4

If we want to express a particular distance, velocity, magnetic field, etc., etc. in a particular direction, we have to combine a magnitude and direction (with units!)

For example, the displacement s is 14 meters in the +x direction:

Or, the magnetic field, “B”, is has a strength of 0.073 teslas in the -y direction:

Or, the nuclear bomber is 8 miles above the Earth’s surface:

Unit vectors for oblique vectors

jiB ˆ5ˆ1

If a vector doesn’t lie conveniently along a particular direction, it can be expressed as a sum of vectors that do lie along independent directions.

If a vector lies 1 unit along the +x direction and 5 units along the +y direction, it would be expressed as:

51 yx BandBThis contains the same information as:

69.78@099.5BOr:

Example problem Chapter 3 #10 (p. 87)

Adding via components

Unit vectors are orthogonal

kjikjiji ˆ4ˆ2ˆ9ˆ4ˆ3ˆ8ˆ5ˆ1

Orthogonal, or perpendicular in a multidimensional sense, means that x, y and z are completely independent directions.

So, “i” terms can’t mix, algebraically, with “j” or “k” terms:

!!!!!!!!15ˆ4ˆ3ˆ8ˆ5ˆ1 kjiji

Using Vectors

Using Vectors

Example problem Chapter 3 #12 (p. 87)

PRSClicker

Questions

Which figure shows ? A A A1 2 3

(Assume A 3 has twice the magnitude of A1 and A2 . )

Which figure shows 2 − ?A

B

What are the x- and y-components Cx and Cy of vector ?

C

A. Cx = 1 cm, Cy = –1 cmB. Cx = –3 cm, Cy = 1 cmC. Cx = –2 cm, Cy = 1 cmD. Cx = –4 cm, Cy = 2 cmE. Cx = –3 cm, Cy = –1 cm

A. tan–1(Cy /Cx)

B. tan–1(Cx /|Cy|)C. tan–1(Cy /|Cx|)D. tan–1(Cx /Cy)E. tan–1(|Cx |/|Cy|)

Angle φ that specifies the direction of is given by

C